Question

Obtain an expression for acceleration of the particle performing circular motion.

Answer 1

Let a particle is performing circular motion in a circular path of radius r, particle makes $\theta$ with centre of the circular path.

Let, angular velocity is $\omega$, angular acceleration is $\alpha$.

we know, $v=r\omega$

so, $\omega =\frac{v}{r}$

angular acceleration is the rate of change of angular velocity. e.g., $\alpha =\frac{△\omega }{△t}$

or $\alpha =\frac{d\omega }{dt}$

$\alpha =\frac{d\left(\frac{v}{r}\right)}{dt}$

$\alpha =\frac{dv}{dt}\frac{1}{r}$

$\alpha =\frac{a}{r}$

or $a=r\alpha$.....................(i)

Now consider the following diagram,

Here, $\delta \theta =\frac{RQ}{BR}=\frac{dv}{v}$

or $dv=v\delta \theta$

Acceleration, $a=\frac{dv}{dt}$

$a=\frac{v\delta \theta }{dt}$

$a=v\omega$

We know, $v=r\omega$

$a=v\frac{v}{r}$

$a=\frac{v^2}{r}$.